According to the classification of the helicoid shape of thread teeth on worm shaft bodies, the known worms in the prior art can mainly be classified as involute helicoid worms, prolate-involute helicoid worms and Archimedes worms, etc. Both the tip surface (outer envelope) and the reference circle (tooth flank) of this type of worms are a cylinder. Moreover, in recent years there appear K-form cylindrical worms and hollow flank worms, etc. They are all constructed as cylindrical worms. They have the advantages of a lower heat value (i.e., heat generated) and lower requirements for the assembly and adjustment of the center distance and axial position because of a lower number of teeth which are simultaneously meshing, but they have the disadvantages of lower loading capacity and lower transmission efficiency. Therefore, toroidal worm and worm gear transmissions have been proposed which also consist of a shaft body with thread teeth. The tip surface and reference circle of the toroidal worm are not a cylinder, but a concave circular-arc revolution which is formed by a sector of a concave circular-arc rotating around the worm axis. The tip surface therefore has a concave face as disclosed, for example, in the Japanese Patents Nos. 7-42813 and 5-196115. As compared with normal cylindrical worm transmissions, the toroidal worm transmissions have the advantages of less wear, higher transmission efficiency, more simultaneously-meshing teeth and larger loading capacity as well as the ability to bear impact loads. However, these toroidal worm transmissions have some disadvantages such as a greater meshing heat value, the requirement of a housing with a larger volume and more lubricating oil volume as well as more radiating (cooling) ribs. The service life of the associated bearings becomes shorter due to a higher oil temperature. The worm gear sets are sensitive to center distance and axial assembly position. Therefore, they require higher accuracy in center distance and axial position, thus increasing the manufacturing cost.